Biologically based models of carcinogenesis are important and useful for cancer prevention and control and for cancer risk assessment of environmental agents. Because in many practical situations, the process of carcinogenesis is very complex far beyond the traditional MVK two stage model, the traditional approaches are not adequate and are too complicated to be of much use. To handle these problems, in this proposal we propose new innovative approaches. These new innovative procedures are based on state space models, stochastic difference or differential equations and the multi-level Gibbs sampling method. In this pilot study we will use these new innovative procedures to solve some problems of interest in carcinogenesis and to analyze data from NCI SEER projects and data from EPA experiments. Because our methods are sufficiently general, they can be used to solve many other important problems as well. (One problem is the derivation of multivariate models which we will pursue in our future research.) The objectives of this proposal are: (1) To propose new innovative and unified modeling approach for developing carcinogenesis models for cancer prevention and for risk assessment. We will apply these procedures to the EPA data and to the NCI SEER data. (2) To develop biologically supported stochastic models and state space models for colon cancer in US. We will apply these procedures to develop multiple-pathways models for colon cancer and to analyze NCI SEER data. (3) To develop statistical procedures to estimate unknown parameters such as the birth rates, the death rates and the mutation rates of initiated cells in carcinogenesis. We will also develop procedures to predict and estimate the state variables to validate the model and to predict future cancer cases.